几类Keller-Segel趋化性模型的稳态解及其定性性质

Steady states and their qualitative properties of several classes of Keller-Segel models

趋化性是指细菌微生物对环境中某些具有刺激性的化学物质产生的定向趋近或者远离的运动.这种与生俱来的能力对生物趋利避害和适应环境等具有重要的作用.为了研究趋化性运动,Evelyn Fox Keller和Lee Aaron Segel于20世纪70年代初提出了一类抛物型偏微分方程系统用以描述细菌种群密度以及化学物质浓度的动态演化.这类系统尽管结构相对简单,但是其动力学性质丰富,并且能够很好地模拟细菌聚集这一趋化性运动中最重要的现象,因此近年来成为偏微分方程研究领域的热点之一.细菌聚集现象可以通过抛物方程组有限时间爆破解所产生的δ-函数进行数学模拟,也可以通过如尖峰解、内边界层解等具有鲜明空间集中特征的稳态解来模拟.前者已有多个综述文献对之进行详细介绍,而关于后者目前还没有任何综述文章,因此本文旨在填补这个空白并综述几类Keller-Segel型模型的稳态解(特别是尖峰解和内边界层解)及其定性性质.本文将着重介绍关于这类问题研究的经典结果和最新进展,以及这些结果研究中所发展出来的变分法、摄动理论和分叉理论等具有创造性的数学.本文还将介绍这类稳态解系统研究中一些未解决的问题.

Chemotaxis is the directed movement of bacterial organisms towards or away from stimulating chemicals in their environment.This intrinsic ability is important for species to profit from the good,avoid the harm and to survive their environment.In order to study cellular chemotactic movement,Evelyn Fox Keller and Lee Aaron Segel proposed a class of parabolic PDE systems to describe the evolution of bacterial population density and chemical concentration.Though these systemns have relatively simple structures,they admit rich spatial-temporal dynamics and can well be used to model cell aggregation,one of the most important phenomena in chemotaxis,hence become one of the most extensively studied topics in PDEs over the past few decades.Cell aggregation can be modeled by the 8-profile out of finite-time blowups in the parabolic systems,and also by the steady states with striking spatial concentrating profiles such as spikes,transition layers,etc.The former approach has been surveyed by a few articles,while a review of the latter is not available,and therefore it is the goal of this paper to survey the steady states and their qualitative properties of several classes of Keller Segel models,with a focus on the spikes and transition layers.In particular,we shall describe the classical results and recent developments of this problem,as well as inspiring mathematics developed meanwhile such as the variation method,singular perturbation,bifurcation theory.We shall also introduce some open problems concerning stationary systems.